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Mathematics QUESTION #1314

The Fourier series of a function $f(x)$ on $(-\pi,\pi)$ is given by $f(x)=\dfrac{a_0}{2}+\sum_{n=1}^\infty(a_n\cos nx+b_n\sin nx)$. The coefficient $a_0$ is:

$\dfrac{1}{\pi}\int_{-\pi}^{\pi}f(x)\,dx$✔️
$\dfrac{2}{\pi}\int_{-\pi}^{\pi}f(x)\,dx$
$\int_{-\pi}^{\pi}f(x)\,dx$
$\dfrac{1}{2\pi}\int_{-\pi}^{\pi}f(x)\,dx$

Correct Answer Logic:

$a_0=\dfrac{1}{\pi}\int_{-\pi}^{\pi}f(x)\,dx$. Then $a_n=\dfrac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos nx\,dx$ and $b_n=\dfrac{1}{\pi}\int_{-\pi}^{\pi}f(x)\sin nx\,dx$. The constant term in the series is $\dfrac{a_0}{2}$.

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Physics QUESTION #1152

Question 962

The energy of a simple harmonic motion depends upon:

Mass
Frequency
Both Mass and Frequency✔️
None of these

Correct Answer Logic:

The total energy of SHM is given by $E = \frac{1}{2}m\omega^2 A^2 = 2\pi^2 m f^2 A^2$, where $m$ is mass, $f$ is frequency, and $A$ is amplitude. So energy depends on both mass and frequency.

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Mathematics QUESTION #6934

Question 963

From a collection of 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are chosen and arranged in a row such that the dictionary always occupies the middle position. How many such arrangements exist?

At least 500 but less than 750
At least 750 but less than 1000
At least 1000✔️
Less than 500

Correct Answer Logic:

Choose 4 novels from 6: $\binom{6}{4} = 15$ ways.

Choose 1 dictionary from 3: $\binom{3}{1} = 3$ ways.

Arrange 5 books in a row with dictionary fixed in the middle: the 4 novels occupy the 4 remaining positions, giving $4! = 24$ arrangements.

Total $= 15 \times 3 \times 24 = \mathbf{1080}$

Since $1080 \geq 1000$, the answer is at least 1000.

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General Abilities QUESTION #2881

Question 964

Divide $\$120$ between X and Y in the ratio $3:7$. How much does Y get?

$\$36$
$\$84$✔️
$\$40$
$\$60$

Correct Answer Logic:

Sum of parts $= 10$. One part $= 120 / 10 = 12$. Y gets $7 \times 12 = 84$.

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English QUESTION #2781

Question 965

What does the PHRASAL VERB 'BREAK DOWN' mean?

To demolish a building forcefully
To stop functioning / to lose emotional control✔️
To reduce the price of something
To separate from a group

Correct Answer Logic:

'Break down' is a phrasal verb with two main meanings: (1) a machine stops working, or (2) a person loses emotional control and cries. Example: 'The car broke down on the highway.'

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Mathematics QUESTION #6966

Question 966

If $\cos^{-1}x - \cos^{-1}\!\dfrac{y}{2}=\alpha$ where $-1\leq x\leq1,\ -2\leq y\leq2,\ x\leq\dfrac{y}{2}$, then $4x^2-4xy\cos\alpha+y^2$ equals:

$4\sin^2\alpha$✔️
$2\sin^2\alpha$
$4\sin^2\alpha-2x^2y^2$
$4\cos^2\alpha+2x^2y^2$

Correct Answer Logic:

Let $\cos^{-1}x=A$ and $\cos^{-1}(y/2)=B$, so $A-B=\alpha$.

Then $\cos A=x,\ \cos B=y/2,\ \sin A=\sqrt{1-x^2},\ \sin B=\sqrt{1-y^2/4}$.

Using $\cos\alpha=\cos(A-B)=\cos A\cos B+\sin A\sin B=\dfrac{xy}{2}+\sqrt{1-x^2}\sqrt{1-\frac{y^2}{4}}$

$4x^2-4xy\cos\alpha+y^2 = (2x-y\cos\alpha)^2+y^2(1-\cos^2\alpha)$... expanding directly: $= 4x^2+y^2-4xy\cos\alpha$.

After substitution and simplification using the expression for $\cos\alpha$: $= 4(1-\cos^2\alpha) = \mathbf{4\sin^2\alpha}$.

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ETAG General Aptitude QUESTION #3762

Question 967

How many distinct three-digit integers can be formed using three different prime numbers between 0 and 9?

64
24✔️
12
4

Correct Answer Logic:

The prime numbers are {2, 3, 5, 7}. We choose 3 primes in $4C3 = 4$ ways and arrange them in $3! = 6$ ways. Total = $4 \times 6 = 24$.

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Reading and Writing (Module-1) QUESTION #5408

Question 968

Several artworks found among the ruins of the ancient Roman city of Pompeii depict a female figure fishing with a cupid nearby. Some scholars have asserted that the figure is the goddess Venus, since she is known to have been linked with cupids in Roman culture, but University of Leicester archaeologist Carla Brain suggests that cupids may have also been associated with fishing generally. The fact that a cupid is shown near the female figure, therefore, _______. Which choice most logically completes the text?

is not conclusive evidence that the figure is Venus.✔️
suggests that Venus was often depicted fishing.
eliminates the possibility that the figure is Venus.
would be difficult to account for if the figure is not Venus.

Correct Answer Logic:

Since cupids may have been associated with fishing itself (not just Venus), the cupid's presence doesn't conclusively prove the figure is Venus.

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Civil Procedure QUESTION #3748

Question 969

A law that treats people differently based on their 'alienage' (status as a non-citizen) is generally subject to which level of judicial review when passed by a STATE?

Rational Basis.
Intermediate Scrutiny.
Strict Scrutiny.✔️
Clear and Convincing Scrutiny.

Correct Answer Logic:

State laws that discriminate based on alienage are generally subject to strict scrutiny (unless it involves 'self-government' functions like voting or being a police officer). Note: Federal laws on alienage get rational basis.

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Chemistry QUESTION #1011

Question 970

Match List-I (Partial Derivatives) with List-II (Thermodynamic Quantities): (A) $\left(\dfrac{\partial G}{\partial T}\right)_P$ (B) $\left(\dfrac{\partial H}{\partial T}\right)_P$ (C) $\left(\dfrac{\partial G}{\partial P}\right)_T$ (D) $\left(\dfrac{\partial U}{\partial T}\right)_V$ with (I) $C_P$ (II) $-S$ (III) $C_V$ (IV) $V$

(A)-(II), (B)-(I), (C)-(III), (D)-(IV)
(A)-(I), (B)-(II), (C)-(IV), (D)-(III)
(A)-(II), (B)-(I), (C)-(IV), (D)-(III)✔️
(A)-(II), (B)-(III), (C)-(I), (D)-(IV)

Correct Answer Logic:

Using thermodynamic relations: $\left(\frac{\partial G}{\partial T}\right)_P = -S$ (II). $\left(\frac{\partial H}{\partial T}\right)_P = C_P$ (I). $\left(\frac{\partial G}{\partial P}\right)_T = V$ (IV). $\left(\frac{\partial U}{\partial T}\right)_V = C_V$ (III). Matching: A-II, B-I, C-IV, D-III.

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English QUESTION #2761

Question 971

Which sentence uses the SEMICOLON correctly?

She was tired; but she kept working.
She was tired; she kept working.✔️
She was tired; and she kept working.
She was tired; to keep working.

Correct Answer Logic:

A semicolon is used to join two independent (complete) clauses without a conjunction. 'She was tired; she kept working' correctly joins two complete sentences. A semicolon should not be used before conjunctions like 'but' or 'and'.

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Urdu QUESTION #5052

Question 972

اس شعر میں علمِ بیان کی کون سی صورت آئی ہے؟ بوئے گل لے گئی بیروں چمن راز چمن، کیا قیامت ہے کہ خود پھول ہیں غمازِ چمن

تشبیہ
استعارہ
مجازِ مرسل
کنایہ✔️

Correct Answer Logic:

اس شعر میں کنایہ استعمال ہوا ہے کیونکہ پھولوں کے غماز ہونے کا استعمال کنائی انداز میں ہے۔

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Education QUESTION #6716

Question 973

A teacher observes that students can recite definitions but cannot apply concepts to new situations. This gap MOST likely indicates:

Failure to achieve conceptual understanding and transfer✔️
Adequate learning requiring only practice
Normal learning progression
Poor memorization techniques

Correct Answer Logic:

Recitation without application suggests surface-level memorization rather than deep conceptual understanding. True learning involves relational understanding—seeing connections and applying knowledge flexibly to novel contexts, not just reproducing information.

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Math (Module-1) QUESTION #5464

Question 974

For the linear function $f$, the table shows: $x=0, f(x)=29$; $x=1, f(x)=32$; $x=2, f(x)=35$. Which equation defines $f(x)$?

$f(x) = 3x + 29$✔️
$f(x) = 29x + 32$
$f(x) = 35x + 29$
$f(x) = 32x + 35$

Correct Answer Logic:

The slope is $\frac{32-29}{1-0} = 3$ and the $y$-intercept is $29$ ($f(0)=29$).

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Paper-I (Anatomy/Physiology and Biochemistry/Microbiology) QUESTION #3427

Question 975

'Denaturation' of a protein involves the loss of its:

Primary structure only
Tertiary and Secondary structure (Biological activity)✔️
Amino acids
Color

Correct Answer Logic:

Heat, acids, or heavy metals can cause a protein to unfold and stop functioning.

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Physics QUESTION #4405

Question 976

If the surface charge density of an infinite sheet increases by $25\%$, the electric field intensity:

Increases by $25\%$✔️
Increases by $50\%$
Decreases by $25\%$
Remains the same

Correct Answer Logic:

Electric field due to an infinite sheet: $E = \dfrac{\sigma}{2\varepsilon_0}$. Since $E \propto \sigma$ (direct proportion), if $\sigma$ increases by $25\%$, then $E$ also increases by $25\%$.

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Electrical Engineering QUESTION #2144

Question 977

What is the primary function of a Buchholz Relay in a transformer?

Protection against external short circuits
Protection against internal gas-producing faults✔️
Regulation of secondary voltage
Measurement of core temperature

Correct Answer Logic:

A Buchholz relay is a gas-actuated safety device installed to detect internal faults like insulation breakdown or overheating that produce gas within the oil.

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Paper-I (Anatomy/Physiology and Biochemistry/Microbiology) QUESTION #3305

Question 978

'Antiserum' provides which type of immunity?

Active Natural
Active Artificial
Passive Artificial✔️
Passive Natural

Correct Answer Logic:

Antiserum contains pre-formed antibodies (e.g., anti-venom) for immediate protection.

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Dairy Food Science and Technology QUESTION #5894

Question 979

Which correctly describes the order of operations in a typical HTST pasteurization system?

Raw milk $\rightarrow$ pasteurizer $\rightarrow$ regeneration $\rightarrow$ holding tube $\rightarrow$ cooling $\rightarrow$ packaging
Raw milk $\rightarrow$ regeneration (heating) $\rightarrow$ pasteurizer $\rightarrow$ holding tube $\rightarrow$ cooling $\rightarrow$ packaging
Raw milk $\rightarrow$ filter $\rightarrow$ regeneration (heating) $\rightarrow$ holding tube $\rightarrow$ regeneration (cooling) $\rightarrow$ chiller $\rightarrow$ packaging✔️
Raw milk $\rightarrow$ homogenizer $\rightarrow$ holding tube $\rightarrow$ regeneration $\rightarrow$ pasteurization $\rightarrow$ packaging

Correct Answer Logic:

In HTST pasteurization: raw cold milk enters the regeneration section (heated by outgoing hot pasteurized milk to $\approx 57\text{–}65^{\circ}\text{C}$, saving $\approx 75\%$ energy), then enters the heating section (to $\geq 72^{\circ}\text{C}$), passes through the holding tube ($\geq 15$ seconds), returns through regeneration (cooling) and a chiller before packaging.

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Paper-II (Pharmacognosy and Pharmaceutics-I) QUESTION #2960

Question 980

What is the Latin term used in prescriptions meaning 'Before Meals'?

Post cibum
Ante cibum✔️
Hora somni
Statim

Correct Answer Logic:

'Ante cibum' (a.c.) means before meals; 'Post cibum' (p.c.) means after meals.

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